论文标题
K3地毯在最小的理性表面及其平滑表面
K3 carpets on minimal rational surfaces and their smoothings
论文作者
论文摘要
在本文中，我们研究了K3双重结构，以最小的理性表面$ y $。结果表明，在$ y = \ mathbb {f} _e $上，由$ \ mathbb p^1 $参数化的$ y = \ mathbb {f} _e $上有无限的许多非分类抽象K3双结构，其中许多都是投射的。对于$ y = \ mathbb {p}^2 $，存在一个唯一的非分类抽象K3双重结构，该结构是非标准的（请参阅Arxiv中的Drézet文章：2004.04921）。我们表明，所有投射K3地毯都可以平滑为光滑的K3表面。证明的副产品之一表明，除非$ y $被嵌入到各种最小程度上，否则在$ y $上有无限的许多嵌入式K3地毯结构。此外，我们在$ \ mathbb f_e $上显示的任何嵌入式投影k3地毯都出现，$ e <3 $出现为嵌入式嵌入的平坦限制，其退化为$ 2：1 $形态。其余的没有，但我们仍然证明了平滑结果。我们进一步表明，与$ \ mathbb {f} _e $支撑的投射K3地毯相对应的希尔伯特点，由完整的线性系列嵌入，当时只有$ 0 \ leq e \ leq e \ leq 2 $。相比之下，与$ \ mathbb {p}^2 $支撑的投射K3地毯相对应的希尔伯特点始终很平滑。最近的一篇关于Bangere，Gallego和González的论文的结果表明，在本文中没有更高的维度类似物。
In this article, we study K3 double structures on minimal rational surfaces $Y$. The results show there are infinitely many non-split abstract K3 double structures on $Y = \mathbb{F}_e$ parametrized by $\mathbb P^1$, countably many of which are projective. For $Y = \mathbb{P}^2$ there exist a unique non-split abstract K3 double structure which is non-projective (see Drézet's article in arXiv:2004.04921). We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless $Y$ is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on $Y$. Moreover, we show any embedded projective K3 carpet on $\mathbb F_e$ with $e<3$ arises as a flat limit of embeddings degenerating to $2:1$ morphism. The rest do not, but we still prove the smoothing result. We further show that the Hilbert points corresponding to the projective K3 carpets supported on $\mathbb{F}_e$, embedded by a complete linear series are smooth points if and only if $0\leq e\leq 2$. In contrast, Hilbert points corresponding to projective K3 carpets supported on $\mathbb{P}^2$ and embedded by a complete linear series are always smooth. The results in a recent paper of Bangere, Gallego, and González show that there are no higher dimensional analogues of the results in this article.
